Correcting low-resolution measurements

ABSTRACT

Methods and systems to correct low-resolution measurements corresponding to unobservable high-resolution measurements by introducing variation in the plurality of low-resolution measurements to obtain perturbed values for the low-resolution measurements. The perturbed values have a higher resolution than another resolution of the low-resolution measurements. A distribution test is performed on the perturbed values.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation in part application of U.S.Non-Provisional application Ser. No. 17/531,206 filed Nov. 19, 2021, thecontents of which are herein incorporated by reference for all purposes.

TECHNICAL FIELD

The present invention relates generally to a method, system, andcomputer program product for correcting measurements. More particularly,the present invention relates to a method, system, and computer programproduct for correcting and testing the normality of a plurality ofmeasurements having low resolution.

BACKGROUND

Organizations may gather and examine information from a number ofsources to obtain a complete and accurate picture of a subject.Obtaining the information may allow the organization to answer pertinentquestions, assess outcomes, conduct research and forecast futureprobability and trends.

Maintaining the integrity of research, making educated businessdecisions, and assuring product/device quality may all be bolstered byaccurate data collecting.

SUMMARY

In one aspect, a method is disclosed. The method may include receiving aplurality of low-resolution measurements, the plurality oflow-resolution measurements corresponding to a plurality of unobservablehigh-resolution measurements. Variation may be introduced in theplurality of low-resolution measurements by iteratively computing, untila termination criteria is met, corresponding perturbed values for thelow-resolution measurements. The corresponding perturbed values may havea higher resolution than another resolution of the low-resolutionmeasurements. A distribution test may then be run on final perturbedvalues that remain after said termination criteria is met.

The method may also include performing the variation introduction bycomputing, for each low-resolution measurement, a first interval thatcontains a corresponding unobservable high-resolution measurementcorresponding to said each low-resolution measurement. A randomobservation may be generated, for each low-resolution measurement, froma uniform distribution on a defined interval. Each random observationmay be transformed to be uniform on a second interval that correspondsto a distribution function such as a cumulative distribution function ofthe first interval to obtain corresponding rescaled uniformobservations. The cumulative distribution function may be based ondistribution parameters such as mean and standard deviation of saidlow-resolution measurements. Responsive to the transforming, and usingan inverse of the distribution function, said rescaled uniformobservations may be inverse transformed to obtain the correspondingperturbed values. In particular, the transforming and the inversetransforming may be repeated iteratively using new distributionparameters of the corresponding perturbed values until said terminationcriteria is met. The distribution test may be an Anderson-Darling test.The Anderson-Darling test may test for normality or for non-normality.However other tests such as other empirical distribution function (EDF)statistics tests may be used.

In another aspect, a non-transitory computer readable storage medium isdisclosed. The non-transitory computer readable storage medium storedprogram instructions which, when executed by a processor, causes theprocessor to perform a procedure that includes receiving a plurality oflow-resolution measurements, the plurality of low-resolutionmeasurements corresponding to a plurality of unobservablehigh-resolution measurements, introducing variation in the plurality oflow-resolution measurements by iteratively computing, until atermination criteria is met, corresponding perturbed values for thelow-resolution measurements, said corresponding perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements, and running, responsive to the introducing, a distributiontest on final perturbed values that remain after said terminationcriteria is met.

In yet another aspect, a computer system is disclosed. The computersystem includes at least one processor configured to perform the stepsof receiving a plurality of low-resolution measurements, the pluralityof low-resolution measurements corresponding to a plurality ofunobservable high-resolution measurements, introducing variation in theplurality of low-resolution measurements by iteratively computing, untila termination criteria is met, corresponding perturbed values for thelow-resolution measurements, said corresponding perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements, and running, responsive to the introducing, a distributiontest on final perturbed values that remain after said terminationcriteria is met.

Other technical features may be readily apparent to one skilled in theart from the following figures, descriptions, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

To easily identify the discussion of any particular element or act, themost significant digit or digits in a reference number refer to thefigure number in which that element is first introduced.

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented.

FIG. 2 depicts a block diagram of a data processing system in whichillustrative embodiments may be implemented.

FIG. 3 depicts a block diagram of an application in which illustrativeembodiments may be implemented.

FIG. 4 depicts a method in which illustrative embodiments may beimplemented.

FIG. 5 depicts a method in which illustrative embodiments may beimplemented.

FIG. 6 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 7 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 8 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 9 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 10 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 11 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 12 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 13 depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 14A depicts a device from which measurements are obtained inaccordance with one or more illustrative embodiments.

FIG. 14B depicts a plot in accordance with FIG. 14A.

FIG. 14C depicts a plot in accordance with FIG. 14A.

FIG. 14D depicts a plot in accordance with FIG. 14A.

FIG. 15A depicts a device from which measurements are obtained inaccordance with one or more illustrative embodiments.

FIG. 15B depicts a plot in accordance with FIG. 15A.

FIG. 15C depicts a plot in accordance with FIG. 15A.

FIG. 15D depicts a plot in accordance with FIG. 15A.

FIG. 16 depicts a process in which illustrative embodiments may beimplemented.

FIG. 17 depicts a process in which illustrative embodiments may beimplemented.

FIG. 18 depicts a process in which illustrative embodiments may beimplemented.

FIG. 19A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 19B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 20A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 20B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 21A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 21B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 22A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 22B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 23A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 23B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 24A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 24B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 25A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 25B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 25C depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 26A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 26B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 26C depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 26D depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 27A depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 27B depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 27C depicts a plot illustrating a concept according to one or moreillustrative embodiments.

FIG. 27D depicts a plot illustrating a concept according to one or moreillustrative embodiments.

DETAILED DESCRIPTION

The illustrative embodiments recognize that there is a need to improvethe accuracy of measurements and data in general for further examinationor research. For example, some statistical techniques may requireselecting an appropriate distribution for a plurality ofdata/measurements. The illustrative embodiments recognize that whiledistribution tests for continuous distributions may assume that sampledata are truly continuous, measurement devices may inherently have aresolution limit that may effectively round these measurements andcreate ties in the data. For example, a sample dataset that maytheoretically take any real value over a range of positive numbers, maybe obtained from measurements that are retrieved only to the nearest 10,such that the sample may have only a few distinct values, say {30, 40,50, 60, 70, 80}. While the data measured to greater precision usinghigher precision measurement devices may follow a normal distribution, atest of these relatively low-resolution data may erroneously reject ahypothesis that they follow a normal distribution. This may be observedin many practices including, for example, the estimation of processcapability statistics, demonstration that a product, such as a medicaldevice, meets a specific reliability requirement, such as 95/95confidence and reliability, and prediction of future warranty claims andthe costs associated with them. Further, a manufacturer or researchermay need to show that a set of data is compatible with a normaldistribution (or some other specific type of distribution). However, thevariation in the data may be small relative to the measurement device,e.g. it may give measurements rounded to the nearest integer or tenth ofan integer. A test of normality such as the Anderson-Darling test willtend to reject the hypothesis of normality too often if the data arerounded too much. Presently available systems may be limited toemploying higher resolution measurement devices to repeat measurementsand worse, may not even recognize the insufficiency of the resolution ofdevices used leading to false rejections of distribution assumptionsunder a given hypothesis. Such a manner of distribution testing iserror-prone, time consuming and costly particularly if measurements haveto be repeated. Further, such a manner of distribution testing may beespecially prohibitive for applications involving sensitive data, suchas the testing of medical devices wherein false rejections may bemisleading and even damaging. The illustrative embodiments recognizethat this has been a significant and complex pain-point in the industrywhich has hitherto been unresolved with any viable systems and processeslet alone systems and processes that are applicable across manydistributions, and practical situations.

Additionally, when it is not possible to substantiate the use of aspecific distribution, such as the normal distribution, adistribution-free approach may be necessary. Such nonparametricapproaches may require much larger sample sizes, which may be costprohibitive, particularly when the nature of the measurement isdestructive to the part.

The illustrative embodiments described herein generally relate toadjusting for the erroneous rounding or truncation of data/measurementsby perturbing the data at each value over a relatively wider interval,and applying a defined distribution testing to the perturbed data. Bythis unconventional approach, the measurements may more closelyrepresent what a random sample from the corresponding population maylook like.

Distribution testing may be used to evaluate data distribution and totest data for normality. Many statistical tests may be parametric (i.e.,the tests may assume that the data follows a specific distribution, hasa defined shape, and can be described by a few parameters, such as amean and a standard deviation. Some data distributions include thenormal distribution, (also known as the bell curve) and distributionsthat can be transformed to a normal distribution (such as a lognormaldistribution). In addition, non-normal distributions, such as the gammaand Weibull distributions are available. For a normal distribution, mostof the data concentrations may be near the mean, or average value andthe likelihood of obtaining values away from the mean in eitherdirection may taper off the further the concentration is from the mean.Further, an Anderson-Darling statistic may be used to assess how closelydata/measurements adhere to a certain distribution. The smaller thisstatistic is for a given data set and distribution, the better thedistribution fits the data. The Anderson-Darling statistic, may forexample, be used to determine if data fits the normality assumption fora t-test. A null hypothesis (Ho) for the Anderson-Darling testhypotheses may be: The data follows a normal distribution, whereas analternative hypothesis (H1) for the Anderson-Darling test may be: Thedata does not follow a normal distribution. To determine if the datafollows the normal distribution, an appropriate p-value may be used. Ifthe p-value is less than a predetermined alpha (typically 0.05 or 0.10),the null hypothesis that the data is from a normal distribution may berejected.

However, though the benefits of the distribution testing may be limitedby the resolution of the data, presently available systems do notaddress these needs or provide adequate solutions. The illustrativeembodiments therefore recognize that by strategically reintroducingvariation into the data/measurements, false rejections of thedistribution assumptions may occur at the stated type I error rate, oralpha as described hereinafter.

An embodiment can be implemented as a software and/or hardwareapplication. The application implementing an embodiment can beconfigured as a modification of an existing system, as a separateapplication that operates in conjunction with an existing system, astandalone application, or some combination thereof.

Particularly, some illustrative embodiments provide a method thatobtains a plurality of low-resolution measurements for a test system,the plurality of low-resolution measurements corresponding to aplurality of unobservable high-resolution measurement values. The methodintroduces variation, in the plurality of low-resolution measurements byiteratively computing, until a stability criteria is met, perturbedvalues for the low-resolution measurements, said perturbed values havinga higher resolution than another resolution of the low-resolutionmeasurements. Responsive to the computing, the method runs adistribution test on the perturbed data. In the method eachlow-resolution data may have a corresponding perturbed data value.

In another embodiment, the method obtains the set of low-resolutionmeasurements by measuring the values of a property, using alow-resolution measurement device. The values may be quantitative valuesof the property and the low-resolution measurement device may round,truncate or generally imprecisely and/or inaccurately obtain measurementdata, based on, for example, a low quality of said low-resolutionmeasurement device.

This manner of correcting low-resolution measurements and distributiontesting is unavailable in the presently available methods in thetechnological field of endeavor pertaining to statistical and predictiveanalytical platforms. A method of an embodiment described herein, whenimplemented to execute on a device or data processing system, comprisessubstantial advancement of the computational functionality of thatdevice or data processing system in configuring the performance of apredictive analytic platform.

The illustrative embodiments are described with respect to certain typesof machines developing statistical and predictive analytic models basedon data records obtained from low-resolution measurements or data. Theillustrative embodiments are also described with respect to otherscenes, subjects, measurements, devices, data processing systems,environments, components, and applications only as examples. Anyspecific manifestations of these and other similar artifacts are notintended to be limiting to the invention. Any suitable manifestation ofthese and other similar artifacts can be selected within the scope ofthe illustrative embodiments.

Furthermore, the illustrative embodiments may be implemented withrespect to any type of data, data source, or access to a data sourceover a data network. Any type of data storage device may provide thedata to an embodiment of the invention, either locally at a dataprocessing system or over a data network, within the scope of theinvention. Where an embodiment is described using a mobile device, anytype of data storage device suitable for use with the mobile device mayprovide the data to such embodiment, either locally at the mobile deviceor over a data network, within the scope of the illustrativeembodiments.

The illustrative embodiments are described using specific surveys, code,hardware, algorithms, designs, architectures, protocols, layouts,schematics, and tools only as examples and are not limiting to theillustrative embodiments. Furthermore, the illustrative embodiments aredescribed in some instances using particular software, tools, and dataprocessing environments only as an example for the clarity of thedescription. The illustrative embodiments may be used in conjunctionwith other comparable or similarly purposed structures, systems,applications, or architectures. For example, other comparable devices,structures, systems, applications, or architectures therefor, may beused in conjunction with such embodiment of the invention within thescope of the invention. An illustrative embodiment may be implemented inhardware, software, or a combination thereof.

The examples in this disclosure are used only for the clarity of thedescription and are not limiting to the illustrative embodiments.Additional data, operations, actions, tasks, activities, andmanipulations will be conceivable from this disclosure and the same arecontemplated within the scope of the illustrative embodiments.

Any advantages listed herein are only examples and are not intended tobe limiting to the illustrative embodiments. Additional or differentadvantages may be realized by specific illustrative embodiments.Furthermore, a particular illustrative embodiment may have some, all, ornone of the advantages listed above.

With reference to the figures and in particular with reference to FIG. 1and FIG. 2 , these figures are example diagrams of data processingenvironments in which illustrative embodiments may be implemented. FIG.1 and FIG. 2 are only examples and are not intended to assert or implyany limitation with regard to the environments in which differentembodiments may be implemented. A particular implementation may makemany modifications to the depicted environments based on the followingdescription.

FIG. 1 depicts a block diagram of a network of data processing systemsin which illustrative embodiments may be implemented. Data processingenvironment 100 is a network of computers in which the illustrativeembodiments may be implemented. Data processing environment 100 includesnetwork 102. Network 102 is the medium used to provide communicationslinks between various devices and computers connected together withindata processing environment 100. Network 102 may include connections,such as wire, wireless communication links, or fiber optic cables.

Clients or servers are only example roles of certain data processingsystems connected to network 102 and are not intended to exclude otherconfigurations or roles for these data processing systems. Server 104and server 106 couple to network 102 along with storage unit 108.Software applications may execute on any computer in data processingenvironment 100. Client 110, client 112, client 114 are also coupled tonetwork 102. A data processing system, such as server 104 or server 106,or clients (client 110, client 112, client 114) may contain data and mayhave software applications or software tools executing thereon. Server104 may include one or more GPUs (graphics processing units) fortraining one or more models.

Only as an example, and without implying any limitation to sucharchitecture, FIG. 1 depicts certain components that are usable in anexample implementation of an embodiment. For example, servers andclients are only examples and not to imply a limitation to aclient-server architecture. As another example, an embodiment can bedistributed across several data processing systems and a data network asshown, whereas another embodiment can be implemented on a single dataprocessing system within the scope of the illustrative embodiments. Dataprocessing systems (server 104, server 106, client 110, client 112,client 114) also represent example nodes in a cluster, partitions, andother configurations suitable for implementing an embodiment.

Device 120 is an example of a device described herein. For example,device 120 can take the form of a smartphone, a tablet computer, alaptop computer, client 110 in a stationary or a portable form, awearable computing device, or any other suitable device. Any softwareapplication described as executing in another data processing system inFIG. 1 can be configured to execute in device 120 in a similar manner.Any data or information stored or produced in another data processingsystem in FIG. 1 can be configured to be stored or produced in device120 in a similar manner.

Test engine 126 may execute as part of client application 122, serverapplication 116 or on any data processing system herein. Test engine 126may also execute as a cloud service communicatively coupled to systemservices, hardware resources, or software elements described herein.Database 118 of storage unit 108 stores one or more measurements or datain repositories for computations herein.

Server application 116 implements an embodiment described herein. Serverapplication 116 can use data from storage unit 108 for low-resolutiondata correction and testing. Server application 116 can also obtain datafrom any client for correction and testing. Server application 116 canalso execute in any of data processing systems (server 104 or server106, client 110, client 112, client 114), such as client application 122in client 110 and need not execute in the same system as server 104.

Server 104, server 106, storage unit 108, client 110, client 112, client114, device 120 may couple to network 102 using wired connections,wireless communication protocols, or other suitable data connectivity.Client 110, client 112 and client 114 may be, for example, personalcomputers or network computers.

In the depicted example, server 104 may provide data, such as bootfiles, operating system images, and applications to client 110, client112, and client 114. Client 110, client 112 and client 114 may beclients to server 104 in this example. Client 110, client 112 and client114 or some combination thereof, may include their own data, boot files,operating system images, and applications. Data processing environment100 may include additional servers, clients, and other devices that arenot shown. Server 104 includes a server application 116 that may beconfigured to implement one or more of the functions described hereinfor low-resolution measurement correction in accordance with one or moreembodiments.

Server 106 may include a search engine configured to search measurementsor databases in response to a query with respect to various embodiments.The data processing environment 100 may also include a dedicatedmeasurement system 124 which comprises a test engine 126. The dedicatedmeasurement system 124 may be used for performing measurements ofdefined properties, via special purpose measurement devices 128 such asmedical devices, vision and imaging devices, detectors, transducers,sensors instruments used in measuring physical quantities and attributesof real-world objects and events. The dedicated measurement system 124may also be used to test samples using the test engine 126. Themeasurement system 124 may make decisions about the distributionsmeasurements belong to by performing distribution testing tomeasurements responsive to performing perturbations on low-resolutionmeasurements. For example, it may apply an Anderson-Darling test to themeasurements modified by perturbation techniques described herein, whichmay result in data having ideal statistical properties.

An operator of the measurement system 124 can include individuals,computer applications, and electronic devices. The operators may employthe test engine 126 of the measurement system 124 to make predictions ordecisions. An operator may desire that the test engine 126 performmethods to satisfy a predetermined evaluation criteria. Thus, a new andunique way to perturb data to address rounding and similar measurementissues that is effective, statistically appropriate and much moreaccurate than using the Anderson-Darling statistic and p-values on theraw, unadjusted data may be provided.

The data processing environment 100 may also be the Internet. Network102 may represent a collection of networks and gateways that use theTransmission Control Protocol/Internet Protocol (TCP/IP) and otherprotocols to communicate with one another. At the heart of the Internetis a backbone of data communication links between major nodes or hostcomputers, including thousands of commercial, governmental, educational,and other computer systems that route data and messages. Of course, dataprocessing environment 100 also may be implemented as a number ofdifferent types of networks, such as for example, an intranet, a localarea network (LAN), or a wide area network (WAN). FIG. 1 is intended asan example, and not as an architectural limitation for the differentillustrative embodiments.

Among other uses, data processing environment 100 may be used forimplementing a client-server environment in which the illustrativeembodiments may be implemented. A client-server environment enablessoftware applications and data to be distributed across a network suchthat an application functions by using the interactivity between aclient data processing system and a server data processing system. Dataprocessing environment 100 may also employ a service-orientedarchitecture where interoperable software components distributed acrossa network may be packaged together as coherent business applications.Data processing environment 100 may also take the form of a cloud, andemploy a cloud computing model of service delivery for enablingconvenient, on-demand network access to a shared pool of configurablecomputing resources (e.g. networks, network bandwidth, servers,processing, memory, storage, applications, virtual machines, andservices) that can be rapidly provisioned and released with minimalmanagement effort or interaction with a provider of the service.

With reference to FIG. 2 , this figure depicts a block diagram of a dataprocessing system in which illustrative embodiments may be implemented.Data processing system 200 is an example of a computer, such as server104, server 106, or client 110, client 112, client 114, measurementsystem 124 in FIG. 1 , or another type of device in which computerusable program code or instructions implementing the processes may belocated for the illustrative embodiments.

Data processing system 200 is also representative of a data processingsystem or a configuration therein, such as device 120 in FIG. 1 in whichcomputer usable program code or instructions implementing the processesof the illustrative embodiments may be located. Data processing system200 is described as a computer only as an example, without being limitedthereto. Implementations in the form of other devices, such as device120 in FIG. 3 , may modify data processing system 200, such as by addinga touch interface, and even eliminate certain depicted components fromdata processing system 200 without departing from the generaldescription of the operations and functions of data processing system200 described herein.

In the depicted example, data processing system 200 employs a hubarchitecture including North Bridge and memory controller hub (NB/MCH)202 and South Bridge and input/output (I/O) controller hub (SB/ICH) 204.Processing unit 206, main memory 208, and graphics processor 210 arecoupled to North Bridge and memory controller hub (NB/MCH) 202.Processing unit 206 may contain one or more processors and may beimplemented using one or more heterogeneous processor systems.Processing unit 206 may be a multi-core processor. Graphics processor210 may be coupled to North Bridge and memory controller hub (NB/MCH)202 through an accelerated graphics port (AGP) in certainimplementations.

In the depicted example, local area network (LAN) adapter 212 is coupledto South Bridge and input/output (I/O) controller hub (SB/ICH) 204.Audio adapter 216, keyboard and mouse adapter 220, modem 222, read onlymemory (ROM) 224, universal serial bus (USB) and other ports 232, andPCI/PCIe devices 234 are coupled to South Bridge and input/output (I/O)controller hub (SB/ICH) 204 through bus 218. Hard disk drive (HDD) orsolid-state drive (SSD) 226 a and CD-ROM 230 are coupled to South Bridgeand input/output (I/O) controller hub (SB/ICH) 204 through bus 228.PCI/PCIe devices 234 may include, for example, Ethernet adapters, add-incards, and PC cards for notebook computers. PCI uses a card buscontroller, while PCIe does not. Read only memory (ROM) 224 may be, forexample, a flash binary input/output system (BIOS). Hard disk drive(HDD) or solid-state drive (SSD) 226 a and CD-ROM 230 may use, forexample, an integrated drive electronics (IDE), serial advancedtechnology attachment (SATA) interface, or variants such asexternal-SATA (eSATA) and micro-SATA (mSATA). A super I/O (SIO) device236 may be coupled to South Bridge and input/output (I/O) controller hub(SB/ICH) 204 through bus 218.

Memories, such as main memory 208, read only memory (ROM) 224, or flashmemory (not shown), are some examples of computer usable storagedevices. Hard disk drive (HDD) or solid-state drive (SSD) 226 a, CD-ROM230, and other similarly usable devices are some examples of computerusable storage devices including a computer usable storage medium.

An operating system runs on processing unit 206. The operating systemcoordinates and provides control of various components within dataprocessing system 200 in FIG. 2 . The operating system may be acommercially available operating system for any type of computingplatform, including but not limited to server systems, personalcomputers, and mobile devices. An object oriented or other type ofprogramming system may operate in conjunction with the operating systemand provide calls to the operating system from programs or applicationsexecuting on data processing system 200.

Instructions for the operating system, the object-oriented programmingsystem, and applications or programs, such as server application 116 andclient application 122 in FIG. 1 , are located on storage devices, suchas in the form of codes 226 b on Hard disk drive (HDD) or solid-statedrive (SSD) 226 a, and may be loaded into at least one of one or morememories, such as main memory 208, for execution by processing unit 206.The processes of the illustrative embodiments may be performed byprocessing unit 206 using computer implemented instructions, which maybe located in a memory, such as, for example, main memory 208, read onlymemory (ROM) 224, or in one or more peripheral devices.

Furthermore, in one case, code 226 b may be downloaded over network 214a from remote system 214 b, where similar code 214 c is stored on astorage device 214 d in another case, code 226 b may be downloaded overnetwork 214 a to remote system 214 b, where downloaded code 214 c isstored on a storage device 214 d.

The hardware in FIG. 1 and FIG. 2 may vary depending on theimplementation. Other internal hardware or peripheral devices, such asflash memory, equivalent non-volatile memory, or optical disk drives andthe like, may be used in addition to or in place of the hardwaredepicted in FIG. 1 and FIG. 2 . In addition, the processes of theillustrative embodiments may be applied to a multiprocessor dataprocessing system.

In some illustrative examples, data processing system 200 may be apersonal digital assistant (PDA), which is generally configured withflash memory to provide non-volatile memory for storing operating systemfiles and/or user-generated data. A bus system may comprise one or morebuses, such as a system bus, an I/O bus, and a PCI bus. Of course, thebus system may be implemented using any type of communications fabric orarchitecture that provides for a transfer of data between differentcomponents or devices attached to the fabric or architecture.

A communications unit may include one or more devices used to transmitand receive data, such as a modem or a network adapter. A memory may be,for example, main memory 208 or a cache, such as the cache found inNorth Bridge and memory controller hub (NB/MCH) 202. A processing unitmay include one or more processors or CPUs.

The depicted examples in FIG. 1 and FIG. 2 and above-described examplesare not meant to imply architectural limitations. For example, dataprocessing system 200 also may be a tablet computer, laptop computer, ortelephone device in addition to taking the form of a mobile or wearabledevice.

Where a computer or data processing system is described as a virtualmachine, a virtual device, or a virtual component, the virtual machine,virtual device, or the virtual component operates in the manner of dataprocessing system 200 using virtualized manifestation of some or allcomponents depicted in data processing system 200. For example, in avirtual machine, virtual device, or virtual component, processing unit206 is manifested as a virtualized instance of all or some number ofhardware processing units 206 available in a host data processingsystem, main memory 208 is manifested as a virtualized instance of allor some portion of main memory 208 that may be available in the hostdata processing system, and Hard disk drive (HDD) or solid-state drive(SSD) 226 a is manifested as a virtualized instance of all or someportion of Hard disk drive (HDD) or solid-state drive (SSD) 226 a thatmay be available in the host data processing system. The host dataprocessing system in such cases is represented by data processing system200.

With reference to FIG. 3 , this figure depicts a block diagram of anexample configuration for correcting and testing low-resolutionmeasurements. The example embodiment includes application 302. In aparticular embodiment, application 302 is an example of clientapplication 122 or server application 116 of FIG. 1 .

Application 302 receives a set or plurality of low-resolutionmeasurements 306 for a test system. In a particular embodiment, thelow-resolution measurements 306 represents quantitative measurementsobtained by an operator using one or more measurement devices 128. Forexample, the measurements/data may be obtained from manufacturer testingsuch as ISO (International Organization for Standardization) testing ofballoon rated burst pressures, which may enable catheter manufacturersdetermine a rated burst pressure (RBP)—the pressure at which 99.9% ofballoons can survive with 95% confidence. Further, a pin g age is asteel pin used to quickly measure the diameter of a drilled hole inmetal or other material. Pin gages come in sets containing various sizedpins. When measuring hole size, the diameter of the largest pin thatwill fit is recorded as the diameter of the hole. A pin gage measuringsystem may have poor resolution because of the relatively largedifferences in pin gag e diameter from one size gage to the next. Evenfurther, food, beverage, pharmaceutical and medical device manufacturersmay have to carefully seal their product packaging to strictspecifications so the product remains safe for consumption. If the sealis too weak, the packaging may open during shipment. If the seal is toostrong, a consumer may have difficulty opening the packaging. Sealstrength is the maximum force needed to separate the two layers of aseal under particular conditions. Seal strength may be rounded to thenearest Newton per square millimeters, causing low resolution in themeasurements which may make it difficult to assess the true processcapability. In another example, air quality meters, designed to measureair velocity, pressure, gases, temperature, humidity, dust etc. may beused may be used to obtain measurements which may be of low resolution.Of course, these examples are not meant to be limiting as measurementsfrom any continuous distribution may be included.

In the embodiment, interval determination component 304 may beconfigured to determine, based on a resolution of the low-resolutionmeasurements, a first interval known to contain an unobservablehigh-resolution measurement value that corresponds to a low-resolutionmeasurement value. This may be performed for all low-resolutionmeasurement values in a data set. Random observation generationcomponent 308 may generate, for each low-resolution measurement value,random observations from a uniform distribution on an interval (0,1).Data perturbation component 310 may transform, using the transformationcomponent 312, the random observations to be uniform on a secondinterval that is based on a cumulative distribution function of thenormal distribution (or of another distribution being tested) to obtainrescaled uniform observations. The rescaled uniform observations may betransformed back using the inverse cumulative distribution function toobtain perturbed values. This may be repeated under new statistics untila termination criteria is achieved as described hereinafter. Further,the distribution test component 314 may perform a test of whether theperturbed values follow a predefined distribution responsive toobtaining final perturbed values.

FIG. 4 illustrates a process 400 in which illustrative embodiments maybe implemented. The process begins in step 402, wherein process 400obtains a plurality of low-resolution measurements for a test system,the plurality of low-resolution measurements corresponding to aplurality of unobservable high-resolution measurement values. In step404, process 400 introduces variation, in the plurality oflow-resolution measurements by iteratively computing, until a stabilitycriteria is met, perturbed values for the low-resolution measurements,the perturbed values having a higher resolution than another resolutionof the low-resolution measurements. Responsive to final perturbed valuesbeing computed, the process may perform a distribution test on the finalperturbed data.

FIG. 5 illustrates a specific example process 500 the process 400 ofFIG. 4 . The process 500 may begin at step 502, wherein process 500receives a plurality of low-resolution measurements, the plurality oflow-resolution measurements corresponding to a plurality of unobservablehigh-resolution measurements (

, i=1, . . . , n). In an example, “n” number of low-resolutionmeasurements X_(i), i=1, . . . , n may be received. In step 504, process500 computes, for each low-resolution measurement, a first interval[L_(i), H_(i)] that contains a corresponding unobservablehigh-resolution measurement corresponding to the each low-resolutionmeasurement value. The first interval may be based on the range ofpossible values of the unobservable high-resolution data that would havebeen rounded to each observed low-resolution value. E.g. If alow-resolution value 13 is observed, the interval may be 12.5 to 13.5.For a situation where the low-resolution measurements are a roundedversion of the unobservable high-resolution measurement, i.e. where

X i = Δ ⁢ Round ⁢ ( Δ ) ,

i=1, . . . , n the first interval [L_(i), H_(i)] may be obtained asshown:

$\left\lbrack {L_{i},H_{i}} \right\rbrack = \left\lbrack {{X - \frac{\Delta}{2}},{X + \frac{\Delta}{2}}} \right\rbrack$

In step 506, process 500 generates, for each low-resolution measurement,a random observation from a uniform distribution on a defined interval(0,1). Thus, step 506 may generate “n” random observations U_(i), i=1, .. . , n. In step 508, process 500 may estimate the distributionparameters (e.g. mean and standard deviation for the normal distributionand thus sample mean ({circumflex over (μ)}) and sample standarddeviation ({circumflex over (σ)}) for initial estimates) of thelow-resolution measurements. Said sample mean ({circumflex over (μ)})and sample standard deviation ({circumflex over (σ)}) may be estimatedas follows:

$\hat{\mu} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}X_{i}}}$$\hat{\sigma} = \sqrt{\frac{\sum_{i = 1}^{n}\left( {X_{i} - \hat{\mu}} \right)^{2}}{n - 1}}$

In step 510, the process 500 may transform each random observation U_(i)to be uniform on a second interval [F(L_(i)), F(H_(i))], to obtainrescaled uniform observations W_(i), with F being the cumulativedistribution function for the normal distribution with the estimateddistribution parameters (estimated sample mean and estimated samplestandard deviation).

The rescaled uniform observations W_(i) may be computed as follows:

W _(i) ={circumflex over (F)}(L _(i))+({circumflex over (F)}(H_(i))−{circumflex over (F)}(L _(i)))U _(i)

For a normal distribution, the estimated cumulative distributionfunction may be estimated as follows, with ϕ denoting the cumulativedistribution function of the standard normal distribution:

${\hat{F}(x)} = {\Phi\left( \frac{x - \hat{\mu}}{\hat{\sigma}} \right)}$

In step 512, process 500 obtains perturbed values (

, i=1, . . . , n) by inverse transforming the rescaled uniformobservations W_(i), responsive to the transforming step of step 510 andusing an inverse of the cumulative distribution function.

{circumflex over (X)} _(i) ={circumflex over (F)} ⁻¹(W _(i)).

In step 514, process 500 may estimate distribution parameters (e.g. themean and standard deviation in the case of a normal distribution) of theperturbed values. In step 516, process 500 may determine if anevaluation/termination criteria condition is met. The terminationcriteria condition may be whether the standard deviation is stable.Responsive to determining that the termination criteria condition is notmet, process 500 obtains the estimates of step 514 for use, in step 518.In other words, updated estimates of the parameters may be obtainedbased on

. For the normal distribution, these may be the sample mean and samplestandard deviation of these values:

$\hat{\mu} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}{\hat{X}}_{i}}}$$\hat{\sigma} = \sqrt{\frac{\sum_{i = 1}^{n}\left( {{\hat{X}}_{i} - \hat{\mu}} \right)^{2}}{n - 1}}$

In an example, the termination criteria condition is met when apercentage change of the scale parameter/standard deviation is less than0.01%) or until a predetermined maximum number of iterations (e.g., 5)is completed. Further a combination of termination criteria may be used.For example, an updated standard deviation estimate a may be compared toa previous estimate {circumflex over (σ)}_(old) and a termination rule

$\frac{❘{\sigma_{new} - \sigma_{old}}❘}{\sigma_{old}} \leq 0.0001$

used along with performing a maximum of 5 iterations.

Thus, process 500 repeats from step 510 using the new sample mean andthe new sample standard deviation until the termination criteriacondition is met. Upon meeting the termination criteria condition, atest such as the Anderson-Darling normality test may be performed on thefinal perturbed data and process 500 ends thereafter. Of course, theseexamples are not meant to be limiting as variations thereof may beachieved from descriptions herein.

FIG. 6 illustrates example measurements generated to simulate actualmeasurements seen from a medical device. The measurements/data form arandom sample from a normally distributed population. While this data isknown to come from a normally-distributed population, the nullhypothesis that the data is from a normally-distributed population isrejected (P-value 602 of less than 0.005, with 0.05 being a thresholdpass condition). A corresponding Anderson-Darling statistic 604 thatresults (1.258) is high. This failure result for a goodness-of-fit testfor a continuous distribution may consistently occur for measurementssampled from that continuous distribution when the measurements arerounded because of poor measurement resolution.

FIG. 7 illustrates a plot with the same measurements from FIG. 6 beingperturbed according to methods described herein. The null hypothesisthat the measurements are from a normally-distributed population passes(P-value 602 of 0.624, with 0.05 being the threshold pass condition). Acorresponding Anderson-Darling statistic 604 of the perturbed data iscomputed to be low (0.281).

Further, simulation studies were performed that may demonstrate problemsassociated with applying the Anderson-Darling test directly tolow-resolution data, and may further demonstrate the superiorstatistical properties attained by methods described herein. Thesimulations were performed based on sets of 2000 columns of generateddata. In some cases, samples of data from the normal distribution aresimulated. These are cases where the Anderson-Darling test shouldindicate that the normal distribution fits the data well in the vastmajority of cases. Rounding of the data were carried out and varied toachieve different ratios of the rounding to the standard deviation ofthe data. In other cases, the Chi square methods were used to simulatenon-normal data. These are cases where the Anderson-Darling testfrequently is able to show a lack of fit of the normal distribution tothe data. The results were then used to evaluate how the distribution ofp-values for the rounded data with and without perturbation stepsdescribed herein compared to those obtained when the data are notrounded. For rounded data without our method, the distribution ofp-values tends to be too low, which results in rejecting the normaldistribution with too high a probability under the condition where theoriginal data are normally distributed before rounding. This worsens asthe rounding ratio increases. On the other hand, applying perturbationsteps described herein results in the rounded data stabilizing thedistribution of the p-values over a range of practical rounding ratioswhen the original data are from a normal distribution.

More specifically, FIG. 8 shows an example wherein the measurementvalues are generated to follow a normal distribution. The figureillustrates tests for rounded values without perturbation—50 samples802, rounded values without perturbation—100 samples 804, rounded valueswith perturbation—50 samples 806, and rounded values withperturbation—100 samples 808 having defined rounding ratios 810(rounding width/standard deviation). For each defined ratio, the test isrepeated a number of times (e.g., 2000 times) to find the percentage 812of times the test incorrectly rejects the null hypothesis. As can beseen, introducing perturbation provides the rounded values withperturbation—50 samples 806 and the rounded values with perturbation—100samples 808 with significantly less percentage 812 of times the testincorrectly rejects the null hypothesis (y-axis, i.e. proportion oftimes the test made a mistake) than the percentage of times the testincorrectly rejects the null hypothesis for rounded values withoutperturbation—50 samples 802 and rounded values without perturbation—100samples 804. In other words, the more rounding, the worse the valueswithout perturbation performs (approaches 100% with increased rounding),whereas the values with perturbation stays steady throughout therounding.

FIG. 9 illustrates the power (ability to show that the data/measurementis not normal when it truly comes from a non-normal population of theAnderson-Darling test in cases where the data is not generated to followa normal distribution. In this case, the higher the percentage 812 oftimes the null hypothesis that the data follows a normal distribution isrejected, the better. A chi-square distribution with different defineddegrees of freedom 902 and sample sizes 814 used. As can be seen, theperturbation provides sufficient “power” over the range of practicalrounding ratios 810 considered, and power comparable to that ofunrounded highest resolution data initially though it may graduallydecline as the rounding ratio increases. This may be expected with theloss of information imposed by rounding.

FIG. 10 illustrates that the power of the Anderson-Darling test for aT-distribution with 4 degrees of freedom and sample size of 100 may alsohave a power that follows a similar trend as shown in the chi-squareexamples of FIG. 9 . Thus from these examples, it can be seen that themethods herein may provide reduced rejection when the data follows anormal distribution and increased rejection when the data/measurementsdo not follow a normal distribution.

FIG. 11 illustrates how well methods described herein perform or agree(percent agreement 1110) with the performance using full unroundedhigh-resolution data. Rounded values with perturbation—50 samples 1106and rounded values with perturbation—100 samples 1108 provideperformance comparable to having the unrounded high-resolution datawhereas rounded values without perturbation—50 samples 1102 and roundedvalues without perturbation—100 samples 1104 provide poor performancewith increasing rounding. The data were generated to follow a normaldistribution. More specifically, methods described herein may be basedon the idea of reintroducing some of the variation removed from the databy rounding. This may be done by means of randomly generated data. Thus,the result may also be random and may depend on the particular randomdata generated. Therefore, if the method were run again, the result maybe different—possibly very similar, but not necessarily. The methodcarefully achieves this in a way that may preserve the distribution ofthe results one would get if the data were not rounded. This may beperformed in a way that preserves as much of the specific information inthe original data as is available. However, as the rounding ratioincreases, the amount of information available about the complete datamay diminish, and the variability of the results may increase. FIG. 11thus shows how the p-values may relate to the complete-data p-values. Inparticular, FIG. 11 shows how often method discussed herein agreed withthe original-data results concerning whether or not to reject the nullhypothesis or normality.

FIG. 12 shows another example of the percent agreement 1110 wherein thedata follows a non-normal distribution. Both sets of data were generatedbased on a chi-square distribution, but the degrees of freedom arevaried to keep the power roughly the same as the number samples arevaried. For 50 rows, 6 degrees of freedom are used and for 100 rows used12 degrees of freedom are used. It can be seen that tests using roundedvalues with perturbation—50 samples—6 degrees of freedom 1206 androunded values with perturbation—100 samples—12 degrees of freedom 1208performed better than tests using rounded values without perturbation—50samples—6 degrees of freedom 1202 and rounded values withoutperturbation—100 samples—12 degrees of freedom 1204. Moreover, percentagreement 1110 for 1206 and 1208 were still high at about 80% near 0.8rounding and at about 70% with the highest rounding.

FIG. 13 shows another comparison similar to that of FIG. 12 based on a Tdistribution with 4 degrees of freedom with 100 samples each. The figureshows tests using rounded values without perturbation—100 samples—4degrees of freedom 1302 and rounded values with perturbation—100samples—4 degrees of freedom 1304. For even moderately large roundingratios, the 1302 will always reject, and hence all of the agreement isfor sets where the original data also led to rejection of the nullhypothesis. On the other hand, perturbation, as shown by 1304 continuesto be more discriminating, and has better agreement with thefull-resolution data even at the highest rounding ratio considered.

Of course, the examples of FIG. 8 -FIG. 13 are examples and are notmeant to be limiting for the methods described herein as other examplesmay be obtained in light of the descriptions herein.

In more use cases, a bend test for bone plates 1402 of FIG. 14A is shownby FIG. 14B-FIG. 14D. Bone plates are thin metal implants used to holdbone segments in the correct position while they heal after a break orother condition. A bone plate may be attached with screws to align andstabilize a broken bone. A variety of laboratory tests may be performedon bone plates to ensure their safety and efficacy. One such test, astatic bend test, may apply increasing force to the bone plate until itbreaks. Requirements may be set on the minimum force for which 99% ofthe population is expected to survive with 95% confidence. Beforedetermining whether the reliability requirements are met, it may benecessary to determine the underlying distribution of the forcemeasurements. It is common to assume that the forces required to breakthese bone plates follow a normal distribution. If the hypothesis testfor normality concludes that the data is non-normal, the user mayinstead use a non-normal distribution or a distribution-free approach todemonstrate reliability. The resulting force measurements 1404, inNewtons, may be rounded because, for example, the testing machine mayonly apply specific levels of force. For a sample size of N=28, thehistogram of FIG. 14B may indicate that the force measurements 1404 arereasonably symmetric and there isn't much evidence to suggest that theyare non-normal. However, as shown in FIG. 14C, the p-value 602 for theAnderson-Darling Normality test (P=0.013) may reject the null hypothesisof normality and lead to the conclusion that these data are from anon-normal population (with 0.05 being the rejection threshold). Furtherthe Anderson-Darling statistic 604 may be inflated, casting theconclusion of non-normality into doubt. By perturbing the forcemeasurements 1404 into perturbed measurements 1406 a p-value 602 of0.310 is obtained which may indicate that the null hypothesis ofnormality will no longer be rejected, and a normal distribution may beused when computing the necessary reliability estimates.

In another use case, the seal strength for the packaging 1504 of urinarycatheters 1502 may be highly regulated due to the risks associated withimproper packaging. Catheters may be sterilized when packaged so theymay be immediately used upon opening. The seal strength of the catheterpackaging may be tested to ensure that the device remains sterile. Anysection of the seal that is weak or compromised may provide anopportunity for entry of potential contaminants. Seal strength is theforce required to remove the seal from the packaging. Force measurements1506, in pound (force), as shown in FIG. 15B are typically rounded,making it difficult to determine the distribution of the underlyingpopulation. The population distribution may be critical for reportingstatistics such as “Cpk” that determine the capability of themanufacturing process to meet desired specifications for seal strength.As shown in the histogram of FIG. 15B, there may be no evidence tosuggest that these force measurements 1506 are from a non-normalpopulation. However, the p-value 602 for the Anderson-Darling statistic(P-Value=0.019) suggests that the null hypothesis of normality should berejected and concludes that the force measurement 1506 are non-normal.Because the data are rounded, however, the Anderson-Darling statistic604 is inflated (0.919). This may make the measurements appear to befrom a significantly non-normal population when they are not. Byperturbing the measurements to correct the low resolution as shown inFIG. 15D, the p-value 602 of 0.131 suggests that the null hypothesis ofnormality is not inappropriately rejected.

As stated before, non-normal distributions such as the gamma and Weibulldistributions may be also applicable. A Weibull distribution maydescribe the probabilities associated with continuous data. However,unlike a normal distribution, it may also model skewed data allowing itto be versatile to fit a variety of shapes. More specifically, a Weibulldistribution may take the values from other distributions using aparameter “shape parameter”. In some examples, the Weibull distributionmay be used to model time, such as analyzing life data and modelingfailure times or in other cases accessing product reliability. Due todifferences compared to normal distributions, modifications to somemethods described herein may be obtained to handle distributions likethe Weibull distribution. With regards to said modifications, parameterestimation steps may potentially fail when measurements are rounded tozero, since estimates may require taking a logarithm, which is undefinedfor zero. Zero values typically occur when some measurements that fallin a skewed Weibull distribution are rounded. Thus, in a first aspect,for a Weibull distribution, steps described herein may be modified byreplacing “zero” values with small positive numbers as appropriate. In asecond aspect, as described hereinafter, said steps may be performed aplurality of times and median result may be selected based on estimatedshape parameters. In a third aspect, initial estimates of Weibullparameters may be obtained based on the inherent discreteness of themeasurements without a need for multiple iterations. The measurementsherein may be obtained using special purpose measurement devices 128such as medical devices, vision and imaging devices, detectors,transducers, sensors instruments used in measuring physical quantitiesand attributes of real-world objects and events.

FIG. 16 illustrates another specific example process 1600 of the process400 of FIG. 4 adapted for a Weibull distribution. Process 1600 mayhereinafter be referred to as the standard process/method. In this case,the measurements may fit a Weibull distribution. The process 1600 maybegin at step 1602, wherein process 1600 may receive a plurality oflow-resolution measurements 306, the plurality of low-resolutionmeasurements corresponding to a plurality of unobservablehigh-resolution measurements (

, i=1, . . . , n). In an example, “n” number of low-resolutionmeasurements (X_(i), i=1, . . . , n may be received. In step 1604,process 1600 may compute, for each low-resolution measurement value x, afirst interval [L(x), H(x)] with L(x)≤x≤H(x) such that {tilde over(X)}_(i)∈[L(X_(i)), H(X_(i))], i=1, . . . , n. The first interval may bebased on the range of possible values of the unobservablehigh-resolution data that would have been rounded to each observedlow-resolution value. E.g., If a low-resolution value 2 is observed, theinterval may be 1.5 to 2.5. For a situation where the low-resolutionmeasurements 306 are a rounded version of the unobservablehigh-resolution measurements, i.e., where

${X_{i} = {\Delta{Round}\left( \frac{\hat{X_{i}}}{\Delta} \right)}},$

i=1, . . . , n the first interval [L(x), H(x)] may be obtained as shown:

${\left\lbrack {{L(x)},{H(x)}} \right\rbrack = \left\lbrack {{x - \frac{\Delta}{2}},{x + \frac{\Delta}{2}}} \right\rbrack},$

x∈χ.

Herein, the full set of distinct observed low-resolution measurementsmay be denoted by χ and the proportion of the data that equals each ofthese observed values may be denoted by {circumflex over (p)}(x), x∈χ.

The goal may be to test whether the underlying unobservablehigh-resolution measurements come from a Weibull distribution. Thus, thecumulative distribution function of the Weibull distribution with shapeparameter m and scale parameter β may be represented as

${{F\left( {{x;m},\beta} \right)} = {1 - e^{- {(\frac{x}{\beta})}^{m}}}},$

x>0 and its inverse cumulative distribution function may be representedas

${{F^{- 1}\left( {{w;m},\beta} \right)} = {\beta\left\{ {- {\ln\left( {1 - w} \right)}} \right\}^{\frac{1}{m}}}},$

0<w<1.

In step 1606, process 1600 may compute, for each low-resolutionmeasurement, a random observation from a uniform distribution on adefined interval (0,1). Thus, step 1606 may compute “n” randomobservations U_(i), i=1, . . . , n. In step 1608, process 1600 mayestimate the distribution parameters (shape parameter mi and scaleparameter {circumflex over (β)}) of the low-resolution measurements.This may be achieved by applying continuous-data Weibull parameterestimation to the low-resolution measurements, X_(i), i=1, . . . , n,but with a replacement of zero values of the low-resolution measurementswith the value H(0)/2, (the midpoint of the interval of positive valuesthat are rounded to zero).

In step 1610, the process 1600 may transform each random observationU_(i) to obtain rescaled uniform observations W_(i). {circumflex over(F)}(x) is the estimated Weibull cumulative distribution function usingthe estimated parameters, {circumflex over (F)}(x)=F(x;{circumflex over(m)},{circumflex over (β)}). The uniform variables may be transformed asfollows:

W _(i) ={circumflex over (F)}(L(X _(i)))+U _(i)[{circumflex over(F)}(H(X _(i)))−{circumflex over (F)}(L(X _(i)))],i=1, . . . n.

In step 1612, perturbed measurement values may be computed bytransforming the rescaled uniform observations based on the inverse ofthe estimated Weibull cumulative distribution function, {circumflex over(X)}_(i)={circumflex over (F)}⁻¹(W_(i)), i=1, . . . , n.

In step 1614, continuous-data Weibull parameter estimation may beapplied to the perturbed data {circumflex over (X)}_(i), i=1, . . . , nto compute new parameter estimates and the process 1600 may determine instep 1616 whether a termination criterion has been met related to thecomputation of further new parameter estimates. For example, thetermination criterion may be evaluated by comparing the new shapeparameter estimate {circumflex over (m)}_(new) from step 1614 with theprevious estimate {circumflex over (m)}_(old). For example, if thetermination criterion for the change in m is not met and a predeterminedmaximum number of iterations (e.g., 20) has not been reached, step 1618may use the new parameter estimates from step 1614 as the currentparameter estimates to repeat steps 1610-1616. A termination criterionfor the change in m may include evaluating the stability of the shapeparameter computations, i.e., whether

${❘\frac{{\hat{m}}_{old} - {\hat{m}}_{new}}{{\hat{m}}_{new}}❘} \leq {0.000{1.}}$

Once the termination criterion has been met, the Anderson-Darling testfor the Weibull distribution may be applied on the final perturbedvalues {circumflex over (X)}_(i), i=1, . . . , n.

Turning now to FIG. 17 and FIG. 18 , alternative processes ofdetermining whether the low-resolution measurements 306 fit a Weibulldistribution are shown. Specifically, in some cases whereinlow-resolution measurements are skewed around zero, unusually “extreme”values of the shape parameter (relative to the distribution of shapeparameters obtained by applying the method to a particular set oflow-resolution data) may be observed. The randomness of the uniform datamay lead to a distribution rather than a single number. For certaindata, the shape parameters may vary widely and the processes may focuson the center part of the distribution of shape values. “Extreme” valuescan deviate from the center of the distribution in either direction. Itmay be beneficial to avoid extreme values of the shape parameter to getbetter performance of the tests. Accordingly, as shown in FIG. 17 , a“median shape process” 1700 that utilizes the standard method to computeif low-resolution measurements 306 fit a Weibull distribution isdisclosed. The process 1700 may comprise performing, in step 1702 thesubroutine 1622 of FIG. 16 (i.e., steps 1606-1618) using threeindependent uniform samples {U_(j1), . . . , U_(jn)}, j=1, 2, 3. Thismay result in three corresponding sets of final perturbed data when thetermination criterion is met in step 1616. The process 1700 may apply,in step 1704 continuous-data Weibull parameter estimation to each set offinal perturbed data to obtain a predetermined number, (e.g., three) ofshape parameter estimates: {circumflex over (m)}₁, {circumflex over(m)}₂, {circumflex over (m)}₃. It was observed quite surprisingly thatwhile a plurality of other numbers of shape parameter estimates may beused, three produced results that best matched a nominal rejection rateof 5%. The process may then compute, in step 1706 which value of j hasthe median shape parameter {circumflex over (m)}_(j) of the threecomputed. In step 1708, the Anderson-Darling test for a Weibulldistribution may then be performed on the final perturbed data havingthe computed median shape parameter estimate to determine if theunobservable high-resolution measurements underlying the low-resolutionmeasurements 306 belong to a Weibull distribution.

With regards to the three shape parameters, a test was performed whereinprocess 1600 was performed hundred times using independent uniformsamples {U_(j1), . . . , U_(jn)}, j=1, . . . , 100 resulting in hundredsets {{circumflex over (X)}_(j1), . . . , {circumflex over (X)}_(jn)}=1,. . . , 100 of perturbed data for the same low-resolution data. Theestimated Weibull parameters ({circumflex over (m)}_(j),{circumflex over(β)}_(j)) were plotted and identification of which ones had significantAnderson-Darling tests was performed. In cases where the process 1600rejected the null hypothesis above a predetermined high rate, a highertendency to reject among the most extreme values of the parameters wasobserved, especially for low values of the shape parameter used togenerate the data for testing process 1600. Process 1700 attempts toavoid these extreme parameter values by selecting from multiple{({circumflex over (X)}_(j1), . . . , {circumflex over (X)}_(jn)} basedon the median of the {circumflex over (m)}_(j). Furthermore, it wasfound that taking the median of just three shape parameters wassufficient to reduce the rejection rate to the desired level.

Turning now to FIG. 18 , another alternative process that determineswhether the low-resolution measurements 306 fit a Weibull distributionis shown. Herein, a different estimation method for obtaining an initialestimate of the Weibull parameters may be used which may be based on theinherent discreteness of the measurements. The process 1800 may begin atstep 1802 wherein a random sample of Uniform (0, 1) data: U_(i), i=1, .. . , n may be generated. In step 1804, the process 1800 may compute thebest discrete Weibull approximation to the rounded data (low-resolutionmeasurements 306). This may be achieved by computing parameter estimates{circumflex over (m)} and {circumflex over (β)} that

${{minimize}{\sum\limits_{x \in \chi}{{\hat{p}(x)}\ln\left( \frac{\hat{p}(x)}{{F\left( {{{H(x)};m},\beta} \right)} - {F\left( {{{L(x)};m},\beta} \right)}} \right)}}},$

i.e., a maximum likelihood estimate that is based on the actual observedrounded data/low-resolution measurements. Thus, unlike in the previousprocesses, there may be no need for iterations to re-compute parameterestimates. The quantity minimized is the Kullback-Leibler divergencerepresenting the statistical distance of a discretized Weibulldistribution from the observed frequencies of the rounded datameasurements. The minimizing values of the parameters m and p are themaximum likelihood estimates given the rounding of the data. There maybe several ways that the minimizing values may be computed. Agradient-based optimization method may work as there is a closed formfor the cumulative distribution function F. Further, a simplealternating line search for the two parameters may be performed. It maybe helpful to have rough estimates of the parameters to narrow thesearch. One way to obtain such rough estimates may be based onestimating quantiles that have simple relationships to Weibullparameters.

If the α^(th) quantile of the Weibull distribution is denoted asx_(α)=F⁻¹(α;m,β), then

${\beta = {{x_{{0.6}3212}{and}m} = {1/{\ln\left( \frac{x_{{0.6}32121}}{x_{{0.3}07799}} \right)}}}},$

For rounded data, the interval containing the sample quantile may beidentified and then interpolation may be performed. Using logarithmicinterpolation may be desired since it may be consistent undertransformations that take one Weibull distribution into another. Thatis, if the sample α^(th) quantile is known to be in the interval [L(x₀),H(x₀)], then {circumflex over (x)}_(α)=exp{(1−π)ln(L(x₀))+π ln(H(x₀))}may be used, with

$\pi = {\frac{\alpha - {\sum_{x < x_{o}}{\hat{p}(x)}}}{\hat{p}\left( x_{0} \right)}.}$

The rough estimates of the parameters may then

${{be}{\overset{\hat{}}{\beta}}_{rough}} = {{{\overset{\hat{}}{x}}_{0.63212}{and}{\hat{m}}_{rough}} = {1/{{\ln\left( \frac{{\overset{\hat{}}{x}}_{{0.6}32121}}{{\overset{\hat{}}{x}}_{{0.3}07799}} \right)}.}}}$

In step 1806, the uniform variables from step 1802 may be transformed.Herein, let {circumflex over (F)}(x) denote the estimated Weibullcumulative distribution function using the estimated parameters, i.e.,let {circumflex over (F)}(x)=F(x;{circumflex over (m)},{circumflex over(β)}). Process 1800 may use this to transform the uniform variables tovariables W_(i)={circumflex over (F)}(L(X_(i)))+U_(i)[{circumflex over(F)}(H(X_(i)))−{circumflex over (F)}(L(X_(i)))], i=1, . . . n.

Said W variables may then be transformed, in step 1808, using theinverse of the estimated Weibull cumulative distribution function togenerate perturbed data values {circumflex over (X)}_(i)={circumflexover (F)}⁻¹(W_(i)), i=1, . . . , n.

In step 1810, the Anderson-Darling test for the Weibull distribution tothe perturbed data {circumflex over (X)}_(i), i=1, . . . , n from step1808.

These steps described herein constitute significant advancements to theindustry and provide a practical solution of better testinglow-resolution measurements obtained in real world settings to gainknowledge about the distribution the measurements belong to. The stepseliminate many complications inherent in previous solutions for testingof measurements. This may enable companies and industries like medicaldevice industries and laboratories, to better meet the safety andefficiency requirements of regulatory authorities by significantlyreducing erroneous conclusions from hypothesis tests about distributionassumptions leading to better protection for the population as a wholedue to more accurate statistical results. Of course, these examples arenot meant to be limiting as variations thereof may be achieved fromdescriptions herein.

Additionally, some steps described herein may require the application ofWeibull parameter estimation appropriate for continuous data. By lettingY_(i)=−ln(X_(i)), i=1, . . . n and iteratively computing for {circumflexover (θ)} in the equation

${\overset{\hat{}}{\theta} = {{\sum\limits_{j}{Y_{j}/n}} - \frac{\sum_{j}{Y_{j}{\exp\left( {{- Y_{j}}/\overset{\hat{}}{\theta}} \right)}}}{\sum_{j}{\exp\left( {{- Y_{j}}/\overset{\hat{}}{\theta}} \right)}}}},$

the estimate of the Weibull shape parameter may be determined as

$\hat{m} = {\frac{1}{\hat{\theta}}.}$

The estimate of the Weibull scale parameter is may also be determined as

$\hat{\beta} = {\exp{\left\{ {\hat{\theta}{\ln\left\lbrack \frac{\sum_{j}{\exp\left( {{- Y_{j}}/\overset{\hat{}}{\theta}} \right)}}{n} \right\rbrack}} \right\}.}}$

Further, the termination criterion

${❘\frac{{\overset{\hat{}}{m}}_{old} - {\overset{\hat{}}{m}}_{new}}{{\overset{\hat{}}{m}}_{new}}❘} \leq {0.0001}$

is equivalent to

$\frac{❘{\frac{1}{{\overset{\hat{}}{m}}_{new}} - \frac{1}{{\overset{\hat{}}{m}}_{old}}}❘}{\frac{1}{{\overset{\hat{}}{m}}_{old}}} \leq {{0.0}00{1.}}$

The parameter

$\theta = \frac{1}{m}$

is the scale parameter in the extreme-value distribution related to theWeibull distribution by the transformation Y=−ln(X). The family ofextreme-value distributions is a location-scale family, analogous to thenormal distributions. With the scale parameter σ defining thetermination criterion when applying the method to the normaldistribution, the termination criterion can be seen as a natural way totranslate the criterion used in the normal setting.

FIG. 19A-FIG. 24B illustrate evaluation results based on examplemeasurements that follow a particular known distribution and rounded todifferent resolutions, i.e., rounding ratios. These may aid to evaluatethe statistical properties of the processes 1600 (referred to generallyas the “Standard” process), process 1700 (referred to generally as the“Median shape” process) and process 1800 (referred to generally as the“Discrete Weibull” process). For each, 2000 columns of unroundedmeasurements from a known distribution may be provided. Beforeperforming said processes, the unrounded measurements may be roundedusing a rounding width Δ value to achieve a predetermined rounding ratioranging from 0.1 to 1.0. The rounding ratio may be the ratio of therounding width Δ to the standard deviation of the distribution. For eachof the 2000 measurement columns, perturbed measurements may be computedand the result of the Anderson-Darling test for the perturbedmeasurements obtained from the processes. The results may be summarizedby the rejection rate 1902 and the percent agreement 1904 with theunrounded measurement results, the rejection rate being the percentageof the 2000 columns where the Anderson-Darling p-value was less than orequal to 0.05 and the percent agreement being the percentage of the 2000columns where the process agreed with the unrounded measurements, i.e.,both had p-values less than or equal to 0.05 or both had p-valuesgreater than 0.05.

As seen in FIG. 19A-FIG. 24B, the plots show the type of distributionused and its parameters as well as the sample size “n” for each column.The plots also show, in FIGS. 19A, 20A, 21A, 22A, 23A, and 24A, theunrounded measurement rejection rate 1906 obtained when applying theAnderson-Darling test to the unrounded measurements. This may serve as abaseline to judge the other rejection rates against.

In example illustrations, four sets of 2000 Weibull columns may haveshape parameters m=0.8 (FIG. 19A and FIG. 19B), 1.2 (FIG. 20A and FIG.20B), 2 (FIG. 21A and FIG. 21B), and 8 (FIG. 22A and FIG. 22B). Thesemay share the same scale parameter β=5. It can be noted from the plotsthat the statistical properties of the standard process 1600 may be bestsuited for some cases and not necessarily for others. For example, withcomparatively smaller values of shape parameter, e.g., m=0.8 as shown inFIG. 19A and increasing rounding ratios, the rejection rate wasconsiderably higher than the desired 5% rate. The rejection rates forthe standard process 1600 began to coincide more with the desired rateas m increased in FIG. 20A, FIG. 21A and FIG. 22A (1.2, 2 and 8respectively). The Median Shape process 1700 and Discrete Weibullprocess 1800 performed better in FIG. 19A and were comparable to thestandard process in FIG. 20A, FIG. 21A and FIG. 22A. This trend was alsoobserved with the corresponding percent agreement 1110 metric as seen inFIG. 19B, FIG. 20B, FIG. 21B and FIG. 22B wherein the process 1700 and1800 agreed more with the unrounded measurements p-values in FIG. 19Bthan process 1600 did. Thus, the Median Shape process 1700 and theDiscrete Weibull process 1800 may provide the best results inpotentially problematic cases.

Turning now to FIG. 23A to FIG. 24B, two sets of 2000 columns ofnon-Weibull data: a Chi-square distribution with 50 degrees of freedomand a log-normal distribution with location 0 and scale 1 were alsoillustrated. These may allow the investigation of the relative power ofthe three processes. Power may be the rejection rate when the nullhypothesis is false. As seen, the power of all the processes with thelog-normal distribution decreases precipitously as the rounding ratioincreases in FIG. 24A and FIG. 24B though all the processes perform welllow rounding ratios. However, one cannot expect any method to do well athigher rounding ratios and this may be explained by the close similarityof the low-resolution log-normal distribution to that of alow-resolution Weibull distribution. Thus, there may not be enoughinformation contained in highly rounded data to distinguish betweenthese distributions. The power profile for the Chi-square data in FIG.23A and FIG. 23B appears much better (i.e., higher power and higherpercent agreement at the higher rounding ratios than may be seen for thelog-normal data) for all the methods since this distribution continuesto be distinguishable from a Weibull distribution even with considerablerounding.

FIG. 25A-FIG. 27D show further practical applications of the disclosurewherein practical measurements may be taken and perturbed for testing.In a use case as shown in FIG. 25A-FIG. 25C, measurements 2504 may beobtained or generated to simulate actual measurements seen from amedical device. The measurements 2504 may form a random sample from apopulation that follows a Weibull distribution. While the measurements2504 may be known to come from a population that follows a Weibulldistribution, the null hypothesis that the data/measurement is from aWeibull population is incorrectly rejected by the Anderson-Darling test(P-Value<0.010, with α=0.05 being the threshold) as shown in FIG. 25Awhich shows a plot of the measurements on the horizontal axis and thecumulative distribution function of the fitted Weibull distribution onthe vertical axis. For example, the fitted line passes roughly throughData=50 and Percent=30%. This means that the probability is about 0.30of a Weibull variable with shape 9.128, scale 56.84, falling at or below50. This failure result for a goodness-of-fit test for a continuousdistribution may consistently occur for measurements sampled from thatcontinuous distribution when the measurements are rounded because ofpoor measurement resolution.

As shown in FIG. 25B, the same measurements are perturbed to formperturbed measurements 2502 using the median shape process 1700. Here,the Anderson-Darling test correctly failed to reject the null hypothesisthat the data are from a Weibull population (P-Value>0.250, with α=0.05being the threshold.)

As shown in FIG. 25C, the same measurements may alternatively beperturbed using the Discrete Weibull Approximation process 1800. Here,the Anderson-Darling test correctly failed to reject the null hypothesisthat the data are from a Weibull population (P-Value>0.250, with α=0.05being the threshold.)

Turning to FIG. 26A-FIG. 26D measurements taken with a durometer 130 maybe shown. A durometer 130 may measure the hardness of a material such assilicon by measuring the depth of an indentation in the material createdby a specific force applied steadily over a small amount of time. Theillustrative embodiments herein recognize that a silicone manufacturermay measure the durometer of a silicone sealant, for example, two weeksafter application. The durometer measurements are likely to come from aWeibull-distributed population, however the durometer 130 may round themeasurements. To determine whether the silicone sealant is capable ofmeeting customer specifications, a non-normal capability study thatassumes the measurements are from a Weibull population may be performed.For a sample size of N=500, the histogram of FIG. 26A may indicate thatdurometer readings are likely to follow a Weibull distribution.

However, as the probability plot of FIG. 26B indicates, the p-value 602for the Anderson-Darling test for a Weibull distribution (P=0.024)suggests that the durometer readings 2602 are not from a Weibulldistributed population because the p-value 602 is less than thethreshold of α=0.05. By perturbing the durometer readings 2602 intoperturbed durometer readings 2604, a p-value>0.25 may be obtained usingboth the median shape method and the discrete Weibull approximationprocesses as shown in FIG. 26C and FIG. 26D respectively.

Turning now to FIG. 27A-FIG. 27D, the illustrative embodiments showfurther applications with “Dwell-Time” measurements, an indication ofthe amount of time a person may spend on a web page. Said amount of timetends to follow a Weibull distribution with a shape parameter less thanone because the decreasing Weibull hazard is such that the person mayeither not find the information searched for and leave quickly or findthe information useful and remain on the page longer. This metric, knownas dwell time, may be an important metric for online marketers due tothe ability to measure a visitor's engagement with web content.

For an illustrative sample size of N=12830, the histogram of FIG. 27Amay indicate that dwell time readings in minutes are likely to follow aWeibull distribution, however the nature of the data may be such thatmany readings are just above zero and when rounded become exactly zerowhich may make it not possible to fit a log-based distribution such asthe Weibull. Additionally, the measurements may be sufficientlydiscrete, particularly at the low end, that no distribution may be foundto fit. The horizontal offset of the plotted data points in theindividual value plot of FIG. 27B may highlight the discrete nature ofthe measurements. By perturbing the dwell time measurements, ap-value>0.25 may be obtained using both the median shape method and thediscrete Weibull approximation method as shown in FIG. 27C and FIG. 27Drespectively.

Of course, these are merely specific examples and are not meant to belimiting as further examples of practical real world quantitativemeasurements taken with a measurement device such as a special purposemeasurement device or equipment (such as medical devices, vision andimaging devices, detectors, transducers, sensors and instruments used inmeasuring physical quantities and attributes of real-world objects) maybe obtained for computations in light of the specification.

Any specific manifestations of these and other similar example processesare not intended to be limiting to the invention. Any suitablemanifestation of these and other similar example processes can beselected within the scope of the illustrative embodiments.

Thus, a computer implemented method, system or apparatus, and computerprogram product are provided in the illustrative embodiments forcorrecting low-resolution measurements and other related features,functions, or operations. Where an embodiment or a portion thereof isdescribed with respect to a type of device, the computer implementedmethod, system or apparatus, the computer program product, or a portionthereof, are adapted or configured for use with a suitable andcomparable manifestation of that type of device.

Where an embodiment is described as implemented in an application, thedelivery of the application in a Software as a Service (SaaS) model iscontemplated within the scope of the illustrative embodiments. In a SaaSmodel, the capability of the application implementing an embodiment isprovided to a user by executing the application in a cloudinfrastructure. The user can access the application using a variety ofclient devices through a thin client interface such as a web browser, orother light-weight client-applications. The user does not manage orcontrol the underlying cloud infrastructure including the network,servers, operating systems, or the storage of the cloud infrastructure.In some cases, the user may not even manage or control the capabilitiesof the SaaS application. In some other cases, the SaaS implementation ofthe application may permit a possible exception of limited user-specificapplication configuration settings.

The present invention may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, configuration data for integrated circuitry, oreither source code or object code written in any combination of one ormore programming languages, including an object oriented programminglanguage such as Smalltalk, C++, or the like, and procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The computer readable program instructions may executeentirely on a dedicated measurement system 124 or user's computer,partly on the user's computer or measurement system 124 as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer or entirely on the remote computer or server, etc. In thelatter scenario, the remote computer may be connected to the user'scomputer through any type of network, including a local area network(LAN) or a wide area network (WAN), or the connection may be made to anexternal computer (for example, through the Internet using an InternetService Provider). In some embodiments, electronic circuitry including,for example, programmable logic circuitry, field-programmable gatearrays (FPGA), or programmable logic arrays (PLA) may execute thecomputer readable program instructions by utilizing state information ofthe computer readable program instructions to personalize the electroniccircuitry, in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general-purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks may occur out of theorder noted in the Figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

All features disclosed in the specification, including the claims,abstract, and drawings, and all the steps in any method or processdisclosed, may be combined in any combination, except combinations whereat least some of such features and/or steps are mutually exclusive. Eachfeature disclosed in the specification, including the claims, abstract,and drawings, can be replaced by alternative features serving the same,equivalent, or similar purpose, unless expressly stated otherwise.

What is claimed is:
 1. A method comprising: measuring, by alow-resolution measurement device, a plurality of low-resolutionmeasurements, the plurality of low-resolution measurements having a lowresolution relative to a resolution of a corresponding plurality ofunobservable high-resolution measurements; receiving the plurality oflow-resolution measurements; introducing variation to the plurality oflow-resolution measurements by iteratively computing, until atermination criteria is met, corresponding perturbed measurements forthe low-resolution measurements, said corresponding perturbedmeasurements having a higher resolution than another resolution of thelow-resolution measurements, a first final set of correspondingperturbed measurements being obtained after the termination criteria ismet; performing the introducing and running a predefined number of timesto obtain other final sets of corresponding perturbed measurements;identifying the final set of corresponding perturbed measurements havinga median shape parameter; running, a distribution test for a Weibulldistribution on the identified final set of corresponding perturbedmeasurements.
 2. The method of claim 1, wherein the distribution test isan Anderson-Darling test for the Weibull distribution.
 3. The method ofclaim 1, wherein the plurality of low-resolution measurements aredetermined to be from the Weibull distribution.
 4. The method of claim1, wherein the introducing comprises: computing, for each low-resolutionmeasurement, a first interval that contains a corresponding unobservablehigh-resolution measurement corresponding to said each low-resolutionmeasurement; generating, for each low-resolution measurement, a randomobservation from a uniform distribution on a defined interval;estimating distribution parameters of said low-resolution measurementsby applying a continuous-data Weibull parameter estimation to theplurality of low-resolution measurements; transforming each randomobservation to be uniform on a second interval that corresponds to adistribution function of the first interval to obtain correspondingrescaled uniform observations, said distribution function being based onestimated distribution parameters of said low-resolution measurements;and inverse transforming, responsive to the transforming, and using aninverse of the distribution function, said rescaled uniform observationsto obtain said corresponding perturbed values, wherein the transformingand inverse transforming are repeated iteratively using new estimateddistribution parameters of the corresponding perturbed values until saidtermination criteria is met.
 5. The method of claim 4, wherein theestimated distribution parameters are a shape and a scale parameter. 6.The method of claim 4, further comprising: replacing, in the estimating,low resolution measurements having a value of zero with a positivevalue.
 7. The method of claim 6, the positive value is a midpoint of theinterval of positive unobservable high-resolution measurements that arerounded to zero by the low-resolution measurement device.
 8. The methodof claim 1, wherein low-resolution measurement device is a deviceselected from the list consisting of a medical device, a vision/imagingdevice, a detector, a transducer, a sensor and an instrument used inmeasuring physical quantities and attributes of real-world objects andevents.
 9. The method of claim 1, wherein the plurality oflow-resolution measurements are rounded versions of the unobservablehigh-resolution measurements.
 10. The method of claim 1, wherein thetermination criteria is a stable shape parameter condition.
 11. Themethod of claim 1, further comprising: obtaining the plurality oflow-resolution measurements by measuring values of a defined property,using a low-resolution measurement device and wherein the values of thedefined property are quantitative values.
 12. A method comprising:measuring, by a low-resolution measurement device, a plurality oflow-resolution measurements, the plurality of low-resolutionmeasurements having a low resolution relative to a resolution of acorresponding plurality of unobservable high-resolution measurements;receiving the plurality of low-resolution measurements; introducingvariation to the plurality of low-resolution measurements by computingcorresponding perturbed measurements for the low-resolutionmeasurements, said corresponding perturbed measurements having a higherresolution than another resolution of the low-resolution measurements;running, a distribution test for a Weibull distribution on thecorresponding perturbed measurements.
 13. The method of claim 12,wherein the distribution test is an Anderson-Darling test for theWeibull distribution.
 14. The method of claim 12, wherein thecorresponding perturbed measurements are computed based on computingparameter estimates that minimize a Kullback-Leibler divergencerepresenting the statistical distance of a discretized Weibulldistribution from observed frequencies of the low-resolutionmeasurements.
 15. The method of claim 14, further comprising: computing,for each low-resolution measurement, a first interval that contains acorresponding unobservable high-resolution measurement corresponding tosaid each low-resolution measurement; generating, for eachlow-resolution measurement, a random observation from a uniformdistribution on a defined interval; transforming each random observationto be uniform on a second interval that corresponds to a distributionfunction of the first interval to obtain corresponding rescaled uniformobservations, said distribution function being based on the parameterestimates; and inverse transforming, responsive to the transforming, andusing an inverse of the distribution function, said rescaled uniformobservations to obtain said corresponding perturbed measurements. 16.The method of claim 14, the parameter estimates are a shape and a scaleparameter.
 17. The method of claim 12, wherein the low-resolutionmeasurements are rounded versions of the unobservable high-resolutionmeasurements.
 18. The method of claim 12 wherein the distribution testis an Anderson-Darling test for the Weibull distribution.
 19. Anon-transitory computer readable storage medium storing programinstructions which, when executed by a processor, causes the processorto perform a procedure comprising the steps of: receiving a plurality oflow-resolution measurements, the plurality of low-resolutionmeasurements having a low resolution relative to a resolution of acorresponding plurality of unobservable high-resolution measurements;introducing variation to the plurality of low-resolution measurements byiteratively computing, until a termination criteria is met,corresponding perturbed measurements for the low-resolutionmeasurements, said corresponding perturbed measurements having a higherresolution than another resolution of the low-resolution measurements, afirst final set of corresponding perturbed measurements being obtainedafter the termination criteria is met; performing the introducing andrunning a predefined number of times to obtain other final sets ofcorresponding perturbed measurements; identifying the final set ofcorresponding perturbed measurements having a median shape parameter;running, a distribution test for a Weibull distribution on theidentified final set of corresponding perturbed measurements.
 20. Acomputer system comprising: a low-resolution measurement device,configured to measure a plurality of low-resolution measurements, theplurality of low-resolution measurements having a low resolutionrelative to a resolution of a corresponding plurality of unobservablehigh-resolution measurements; and at least one processor configured toperforms the steps of: receiving the plurality of low-resolutionmeasurements, the plurality of low-resolution measurements having a lowresolution relative to a resolution of a corresponding plurality ofunobservable high-resolution measurements; introducing variation to theplurality of low-resolution measurements by iteratively computing, untila termination criteria is met, corresponding perturbed measurements forthe low-resolution measurements, said corresponding perturbedmeasurements having a higher resolution than another resolution of thelow-resolution measurements, a first final set of correspondingperturbed measurements being obtained after the termination criteria ismet; performing the introducing and running a predefined number of timesto obtain other final sets of corresponding perturbed measurements;identifying the final set of corresponding perturbed measurements havinga median shape parameter; running, a distribution test for a Weibulldistribution on the identified final set of corresponding perturbedmeasurements.
 21. A non-transitory computer readable storage mediumstoring program instructions which, when executed by a processor, causesthe processor to perform a procedure comprising the steps of: measuring,by a low-resolution measurement device, a plurality of low-resolutionmeasurements, the plurality of low-resolution measurements having a lowresolution relative to a resolution of a corresponding plurality ofunobservable high-resolution measurements; receiving the plurality oflow-resolution measurements; introducing variation to the plurality oflow-resolution measurements by computing corresponding perturbedmeasurements for the low-resolution measurements, said correspondingperturbed measurements having a higher resolution than anotherresolution of the low-resolution measurements; running, a distributiontest for a Weibull distribution on the corresponding perturbedmeasurements.
 22. A computer system comprising: a low-resolutionmeasurement device, configured to measure a plurality of low-resolutionmeasurements, the plurality of low-resolution measurements having a lowresolution relative to a resolution of a corresponding plurality ofunobservable high-resolution measurements; and at least one processorconfigured to performs the steps of: measuring, by a low-resolutionmeasurement device, a plurality of low-resolution measurements, theplurality of low-resolution measurements having a low resolutionrelative to a resolution of a corresponding plurality of unobservablehigh-resolution measurements; receiving the plurality of low-resolutionmeasurements; introducing variation to the plurality of low-resolutionmeasurements by computing corresponding perturbed measurements for thelow-resolution measurements, said corresponding perturbed measurementshaving a higher resolution than another resolution of the low-resolutionmeasurements; running, a distribution test for a Weibull distribution onthe corresponding perturbed measurements.